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OK I am confused.  I have a cool fractal that has been around since the dawn of time.  It is a recursive function but it is a series.  

I will use Hertz as an example for the frequency.
I will use a Music Bar to represent the frequency bit, the size of the bar stays the same, the amount of notes in the bar can increase or decrease.

1 = 1 Hertz frequency would be the number 1...One Beat Per Bar

2 = 2 Hertz frequency would be the number 2...2 - 1/2 notes

3 = 1 - 2 Hertz Frequency and 2 - 4 Hertz Frequency Notes

4 = 4 - 4 Hertz Notes

5 = 3 - 4 Hertz Notes and 2 - 8 Hertz Notes
6 = 2 - 4 Hertz Notes and 4 - 8 Hertz Notes
7 = 1 - 4 Hertz Note and 6 - 8 Hertz Notes

8 = 8 - 8 Hertz Notes

9 = 7 - 8 Hertz Notes and 2 - 16 Hertz Notes
10 = 6 - 8 Hertz Notes and 4 - 16 Hertz Notes
11 = 5 - 8 Hertz Notes and 6 - 16 Hertz Notes
12 = 4 - 8 Hertz Notes and 8 - 16 Hertz Notes
13 = 3 - 8 Hertz Notes and 10 - 16 Hertz Notes
14 = 2 - 8 Hertz Notes and 12 - 16 Hertz Notes
15 = 1 - 8 Hertz Note and 14 - 16 Hertz Notes

16 = 16 Hertz Notes...

and so on and so on, the pattern continues, the bar size stays the same but the frequency sets increase in side of it.

I would describe the number 12 as 4/8 Hz and 8/16 Hz, meaning there are 4 quick 8 Hz Waves and 8 quicker 16 Hertz Waves...

Do you see the pattern? The numbering system is different than Binary's 1, 2, 4, 8.... and so on...this is a frequency set.

Here is a Chart describing the pattern...

(http://ashesmi.yolasite.com/resources/Frequency%20Set.jpg)




Here is a simple video describing the frequency set  http://www.youtube.com/watch?v=JGozCCN4344 (http://www.youtube.com/watch?v=JGozCCN4344)

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Greetings and Salutations, again outlier, gets a little lonely where you R.

You seek! You seek. You seek, but are U ready? Some answers are not 4 U. Are U ready? We know your not! This is why your puzzle up until now, no one has come to help you! Your  puzzle, stays unsolved, and will stay unsolved. But alone u R not, 4 here student of the Universes, there once was a nice man, who lived so long he lived to see a descendent look like him, that is your clue number one to solve your puzzle!
Clue number two, he fell in love with one of his descendants, and births, there were many! Clue number three 4 U, the day he was schedule to pass away, he smiled into the face of a son, and his son looked like him, and that is all the clues U R allowed, Mr. Kevinmorals, student of the Universes, until next time, Peace!

 .

I am afraid that you should write the function first, before we can help?
;)

I am not sure what the question is but I do see one use for this numbering system. If you only have a piece of rope to count with you say 1 = unfolded, 2=fold in half, 3=fold one of the halves in half, 4=fold the other half in half etc.
Then you can count the folds later on. Possibly shepherds could use this as they walk to keep track of the flock. Other than that - utterly useless ;)
Just kidding but not much of a numbering system without zero or negatives :(

This reminds me Pythagoras' ideas about harmonics and tuning.  You are right when you say this is very ancient.  Check out http://en.wikipedia.org/wiki/Pythagorean_tuning (http://en.wikipedia.org/wiki/Pythagorean_tuning) for a good place to start.